The strong fractional choice number of triangle-free planar graphs

TL;DR

This paper proves that all triangle-free planar graphs have a strong fractional choice number at most 15/4, providing the first non-trivial upper bound for this class and answering a question from prior research.

Contribution

It establishes a new upper bound on the strong fractional choice number for triangle-free planar graphs, advancing understanding of their list coloring properties.

Findings

Every triangle-free planar graph is (15m,4m)-choosable for any positive integer m.

The strong fractional choice number of these graphs is at most 15/4.

This result confirms a conjecture for the case m=1.

Abstract

Let $a,b$ be positive integers with $a\ge b$. A graph $G$ is $(a,b)$-choosable if, for every assignment of lists $L(v)$ of size $a$ to the vertices of $G$, there exists a choice of subsets $C(v)\subseteq L(v)$ with $|C(v)|=b$ for each $v$ such that $C(u)\cap C(v)=\emptyset$ whenever $uv\in E(G)$. We show that every triangle-free planar graph is $(15m,4m)$-choosable for any positive integer $m$. As an immediate consequence, the strong fractional choice number of triangle-free planar graphs is at most $15/4$. This appears to be the first non-trivial upper bound on this parameter for this class of graphs. In particular, the case $m=1$ answers affirmatively a question posed by Jiang and Zhu in [J.~Combin.\ Theory Ser.~B, 2019].